Let $Z_0(G) = \{ 1 \}$ and:
$$Z_{i + 1}(G)/Z_i(G) = Z(G/Z_i(G))$$
(as defined in dummit and foote). I want to show that $Z_i(G)H/H \subseteq Z_i(G/H)$. I can see it's true for $i = 0$ and $i = 1$, but I'm stuck in the inductive case because I have to work with quotients of quotients (and not in the form $(G/H)/(K/H)$ so no 3rd isomorphism theorem).
In other words, I have an element $zH \in Z_{i + 1}(G)H/H$. I want to show that it is in $Z_{i + 1}(G/H)$, which is defined recursively as
$$Z_{i + 1}(G/H)/Z_i(G/H) = Z((G/H)/Z_i(G/H)).$$